The British mathematician Godfrey Harold Hardy famously said, more than any other art or science, mathematics is more of a young man's game. Here, are his comments on young people How correct is the understanding?
Granted, proving a mathematical theorem requires a lot of creativity, the ability to rethink the problem, and to think in ways no one else has thought of. However, it also requires a lot of experience and knowledge reserves. After all, if you don't understand a problem, you can't justify it. Many unproven conjectures are based on a mountain of concepts that often take years to reach the top. Based on Wikipedia, I present a timeline of the development of mathematics from 1501 to 2015, tracing back 250 major events in the field of mathematics: new proofs of theorems, the publication of important work, or the germination of core mathematical concepts.
Here are some of those events:
In 1540, 18-year-old Lodovico Ferrari solved the quartic equation.
In 1799, 22-year-old Carl Friedrich Gauss proved the fundamental theorem of algebra (every polynomial equation has a solution in complex numbers).
In 1925, 24-year-old Werner Heisenberg and Jordan and Born established a matrix representation of quantum mechanics.
In 2004, 29-year-old Terence Tao and Ben Green proved the Green-Tao theorem.
In 1522, 30-year-old Adam Ries explained the use of Arabic numerals and their advantages over Roman numerals.
In 2003, 37-year-old Grigori Perelman proved the Poincaré conjecture.
In 1994, 41-year-old Andrew Wiles proved part of Taniyama Shimura's conjecture, thus proving Fermat's Last Theorem.
In 1929, 47-year-old Emmy Noether first introduced a general representation theory for groups and algebras.
In 2013, 58-year-old Zhang Yitang proved that there are infinitely many pairs of prime numbers with finite gaps.
The base e of the natural logarithm was first mentioned in 1618 by 68-year-old John Napier in a book on logarithms.
Age of great mathematicians
Data summary:
o their average age is 37
o The median age is slightly lower at 35
Ø 25 % of mathematicians in Mathematical Memorabilia have made important mathematical achievements under the age of 30
Ø 42 % of mathematicians made important achievements between the ages of 30-39
Ø 33 % of mathematicians achieved important results at the age of 40 or over
Ø Among them, the youngest is 18 years old, in 1540 Lodovico Ferrari (Lodovico Ferrari) derived the general solution of the quartic equation
Ø The oldest was 73 years old, as demonstrated in 1825 by Adrien-Marie Legendre and Peter Gustav Lejeune Dirichlet together For the case of n=5 Fermat's Last Theorem.
From 20 to 70 years old
We made an interactive chart, The Age of the Great Mathematicians, which readers can click here or below for detailed information (use the sliders to investigate the achievements of mathematicians of all ages). Some of the highlights are worth noting:
Between the ages of 20 and 29
In 1832, the French mathematician Évariste Galois proposed the general conditions for the solvability of algebraic equations when he was 21 years old, so he basically established group theory and Galois theory. A man who used the mathematical term "group" to denote a set of permutations and is credited with Niels Abel as the founder of modern group theory. But what is tragic and legendary is that he was killed in a duel not long after he came up with these theories.
In 1913, the 26-year-old Indian mathematician Srinivasa Ramanujan wrote a letter to Hardy with a long list of unproven theorems. Nugin also pleads with Hardy to help him out of his current state of poverty. Of course, Ramanujan's discovery in the letter must have predated his 26-year-old.
Between the ages of 30 and 39
In 2008, at the age of 31, the Tehran mathematician Marian Mirzakhani proved a long-unsolved conjecture: William Thurston proposed that earthquakes in Teichmüller space Flow (Earthquake map flow) are to traverse the system. Six years later, she was awarded the Fields Medal "for outstanding contributions to the dynamics and geometry of Riemann surfaces and their modulo spaces". On July 14, 2017, Mirzakhani died of breast cancer at the age of 40.
In 1837, the 32-year-old German mathematician Johann Peter Gustav Lejeune Dirichlet developed analytic number theory in a paper on the existence of prime numbers in a given arithmetic series. This is Dirichlet's fourth job on the mathematical timeline, the first three taking place when he was 20, 27, and 32 years old.
1915. The 36-year-old theoretical physicist Albert Eins published his general theory of relativity and his special theory of relativity ten years ago when he was a staff member at the Swiss Patent Office.
Between the ages of 40 and 49
In 1993, after several years of secretly working on Fermat's Last Theorem, 40-year-old Andrew Wiles announced that he had proved Fermat's Last Theorem. It is known that during the review process there was an error in this certificate, but the error was corrected the following year. Wiles expanded his work at the age of 46, completing all of Taniyama Shimura's conjectures.
In 1918, 41-year-old GH Hardy and Srinivasa Ramanujan together developed an asymptotic formula for the division function. Perhaps his collaboration with the young creative genius Ramanujan is one of the reasons why he laments that mathematics is "a game for young people."
50+
In 2013, Chinese-American mathematician Zhang Yitang proved for the first time that there are infinitely many pairs of prime numbers with finite gaps, thus achieving a qualitative breakthrough in the number theory problem of the twin prime number conjecture. In 1991, Zhang Yitang did not get a letter of recommendation from his supervisor after obtaining his doctorate. His academic road was bumpy. For a period of time, he made a living by doing odd jobs. He worked as an accountant for several years and worked in a fast-food restaurant on subway.
In 1722, the French mathematician Abraham de Moivre connected complex numbers with trigonometry and proposed the Moivre formula; at the age of 66, the normal distribution was introduced to approximate the binomial distribution.
There is a humorous anecdote about the aging mathematician:
It is often said that De Morpher, who has always been interested in series, once predicted that he would need to sleep 15 minutes more than the previous day and that he would die when the total sleep reached 24 hours, which was November 27, 1754, which was also the time of his death.
Data and Methods
Most of the data is obtained from the mathematical timeline on Wikipedia, starting from the modern (16th century). It would be nice to start with Archimedes and Hypatia in ancient Greek times, but historical records are imprecise, so here we start with modern. We also augmented the data provided on the Wikipedia timeline with some other well-known results, such as the work of mathematical physicists such as Einstein, Bohr, and Heisenberg.
Surprisingly, some of the big recent events in mathematics have also had some data collection problems. We would like to include Hao Huang's 2019 proof of the sensitivity conjecture for Boolean functions, but we can't find his date of birth. Extrapolating from his education, he may be in his 30s.
Also, we like to include Joan Taylor, an amateur mathematician who, along with Joshua Socolar, discovered Socolar-Taylor ) tile collage, solving the problem posed by Roger Penrose. We can't find her date of birth, but on her website, she says she's been thinking about this since 1990, 20 years before the results were published in 2011.
Statistically speaking, while most high-profile achievements are made by mathematicians between the ages of 20 and 40, that age range is still wide. As mathematicians get older, mathematicians tend to gravitate towards book and synopsis type of work, and of course, there are still some examples of groundbreaking innovative proofs made by older mathematicians.
Every mathematician's story is different. Among them, there are prodigies who died young, prolific scholars who cover all fields, and people who insist on studying a problem for 20 years. For those of us amateur mathematicians, we may not want to change the world, but just enjoy knowing it, and it is good enough to know that our brains are always thinking and that we can continue to enjoy mathematics.